On the solution set of additive and multiplicative congruences modulo primes

Abstract

Let p be an odd prime. In this paper, analogs of Wilson?s and Wolstenholme?s theorems on the solution sets S+ = {n ? Z* p | n ? a + b ? ab (mod p)} and S? = {n ? Z* p | n ? a ? b ? ab (mod p)} are given, where Z* p denote a reduced residue system modulo p. We also establish congruences about sum and product of the quadratic residues in S+ or in S? modulo p. Finally, we raise a problem on how to solve Hadamard?s conjecture in the last section.